DE102014209793B4 - Method and device for operating an internal combustion engine - Google Patents

Abstract

Method for operating an internal combustion engine comprising an intake tract (1) and one or more cylinders (Z1 to Z4), to which gas inlet valves (12) and gas outlet valves (13) are assigned, wherein gas exchange valves include gas inlet valves (12) and gas outlet valves (13) in which a model temperature of a gas in the intake tract (1) is determined cyclically for a current point in time in a first operating state, depending on a predefined intake manifold model and free of a measured temperature value of the gas which is assigned to the current point in time, the model temperature for the current point in time is determined depending on a model temperature, which was determined for a previous point in time, - depending on the model temperature determined for the current point in time, a cylinder air mass is determined which, after the gas exchange valves are closed, is in the respective cylinder, in which in a second operating state Tempe The measured temperature value of the gas is provided, which is representative of a temperature of the gas at the current time, - a temperature correction value is determined depending on the model temperature for the current time and the temperature measurement value provided, - the temperature correction value is assigned to the intake manifold model and - at least in the first and the second operating state, the model temperature for the current state is determined as a function of the temperature correction value by means of the intake manifold model and in which the model temperature for the current time is adapted as a function of the temperature measurement value provided, by correcting the model temperature by a predetermined factor in the direction of the temperature measurement value.

Description

The invention relates to a method and a device for operating an internal combustion engine with one or more cylinders, each of which gas inlet valves are assigned.

Ever stricter legal regulations with regard to permissible pollutant emissions from motor vehicles in which internal combustion engines are arranged make it necessary to keep the pollutant emissions as low as possible during the operation of the internal combustion engine. On the one hand, this can be done by reducing the pollutant emissions that arise during the combustion of the air / fuel mixture in the respective cylinders of the internal combustion engine. On the other hand, exhaust gas aftertreatment systems are used in internal combustion engines, which convert the pollutant emissions that are generated in the respective cylinders during the combustion process of the air / fuel mixture into harmless substances. For this purpose, catalytic converters are used that convert carbon monoxide, hydrocarbons and nitrogen oxides into harmless substances.

Both the targeted influencing of the generation of the pollutant emissions during the combustion in the respective cylinder and the conversion of the pollutant components with a high efficiency by the exhaust gas catalytic converter require a very precisely set air / fuel ratio in the respective cylinder.

An intake manifold model is, for example, in the specialist book "Manual Combustion Engine, Fundamentals, Components, Systems, Perspectives", publisher Richard van Basshuysen / Fred Schäfer, 2nd improved edition, June 2002, Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig / Wiesbaden, pages 557 to 559. Such intake manifold models are also in EP 0820559 B1 and EP 0886725 B1 described.

In the DE 11 2010 004 825 T5 describes an engine control device that can accurately estimate intake pipe temperature behavior during a transitional period even in an engine embedded with a variable valve or turbocharger. The engine control device estimates the transition behavior of the intake pipe temperature based on a flow rate of gas flowing into the intake pipe, a flow rate of gas flowing out of the intake pipe, an intake pipe pressure, and a time change rate of the intake pipe pressure. The device performs knock control based on the estimated transition behavior of the intake pipe temperature during the transition period.

In the publication by F. Karlsson: "Modeling the Intake Manifold Dynamics in a Diesel Engine", Linköping, April 2, 2001, - Exam work models for the suction section of internal combustion engines are described. It is disclosed in this publication that not only pressure and gas mass can be used as the state variable, but also pressure and temperature. Using the differential equations for the "pT state model", the model sizes from the previous calculation step can be used to calculate the model temperature and model pressure.

The object on which the invention is based is to create a method and a device for operating an internal combustion engine which makes a contribution to reliable and low-emission operation of the internal combustion engine.

The object is achieved for a method by the features of the independent claims 1 and 2 and for a device by the features of the independent claim 8. Advantageous embodiments are characterized in the subclaims.

The invention is characterized on the one hand by a method and on the other hand by a corresponding device for operating an internal combustion engine with an intake tract and one or more cylinders, each of which gas inlet valves and gas outlet valves are assigned, gas exchange valves comprising gas inlet valves and gas outlet valves.

In a first operating state, a model temperature of a gas in the intake tract is determined cyclically for a current point in time depending on a predefined intake manifold model and free of a temperature measurement value of the gas that is assigned to the current point in time. The model temperature is determined for the current point in time depending on a model temperature for a previous one Time was determined. Depending on the model temperature determined for the current point in time, a cylinder air mass is determined which is in the respective cylinder after the gas exchange valves have been closed.

In a second operating state, a temperature measurement value of the gas is provided which is representative of a temperature of the gas at the current time.
A temperature correction value is determined depending on the model temperature for the current time and the temperature measurement value provided.

The temperature correction value is assigned to the intake manifold model and, at least in the first and the second operating state, the model temperature for the current state is determined depending on the temperature correction value by means of the intake manifold model, and the model temperature is adapted for the current time depending on the temperature measurement value provided by the model temperature is corrected by a predetermined factor in the direction of the temperature measured value.

The first operating state is in particular an unsteady operating state. The previous point in time is particularly assigned to the last cycle.

A temperature sensor in the intake tract often has a relatively large delay. By determining the cylinder air mass free of a temperature measurement value that is assigned to the current point in time, a cylinder air mass can be determined very quickly, thereby making a contribution to reliable and low-emission operation of the internal combustion engine, since the cylinder air mass can be used as the basis for the fuel metering ,

A temperature correction value is determined depending on the model temperature for the current point in time and the temperature measurement value provided. The temperature correction value is assigned to the intake manifold model and at least in the first and the second operating state, the model temperature for the current state is determined as a function of the temperature correction value by means of the intake manifold model.

The second operating state is in particular a quasi-stationary operating state. The quasi-stationary operating state is distinguished, for example, by the fact that all input signals of the intake manifold model are essentially constant for a predetermined time, for example several seconds. Since the temperature of the gas essentially does not change in the second operating state, the temperature measurement value of the gas that is representative of a temperature of the gas at the current time is, for example, the temperature measurement value of the gas that is assigned to the current time or a temperature measurement value of the Gases assigned to the previous point in time.

The temperature correction value is determined, for example, in such a way that the difference between the model temperature and the temperature measured value is minimized. For example, the model variable “temperature of the throttle valve mass flow” of the intake manifold model is corrected using the temperature correction value. As an alternative or in addition, it is also possible to introduce an additional, non-physically modeled model input “heat flow through the intake manifold wall”, which is corrected using the temperature correction value in such a way that the difference between the model temperature and the temperature measured value is minimized. In this way, the cylinder air mass can be determined particularly precisely.

Furthermore, in the second operating state, the temperature measurement value of the gas is provided, which is representative of a temperature of the gas at the current time, and the model temperature is adapted for the current time depending on the temperature measurement value provided.

In the second operating state, the relatively large delay of the temperature sensor may not play a role, since the values of the sensor essentially do not change. In the second operating state, the model temperature can thus be easily adapted to the temperature measured value. This adaptation can in turn be used when changing to the first operating state, since in the first operating state the model temperature for the current point in time is determined as a function of a model temperature that was determined for a previous point in time. As a result, the cylinder air mass can be determined particularly precisely and nevertheless very quickly in both operating states.

In this way, the correction of the cylinder air mass can be carried out in a particularly robust and very simple manner, since, for example, very few calculation steps are necessary for the correction.

In this way, the correction of the cylinder air mass can be carried out particularly robustly and very precisely, since the difference is used in a simple manner for the correction.

According to an advantageous embodiment, a model pressure of a gas in the intake tract is determined cyclically for a current point in time depending on the predefined intake manifold model and free of a pressure measurement value of the gas that is assigned to the current point in time. The model print is determined for the current point in time depending on a model print that was determined for a previous point in time. The cylinder air mass is determined depending on the model pressure determined for the current point in time.

A pressure sensor in the intake tract may also have measurement errors. By determining the cylinder air mass free of a pressure measurement value that is assigned to the current point in time, a cylinder air mass can be determined very quickly, thereby making a contribution to reliable and low-emission operation of the internal combustion engine, since the cylinder air mass is used as the basis for the fuel metering can.

According to a further advantageous embodiment, a pressure measurement value of the gas is provided in the second operating state, which is representative of a pressure of the gas at the current time. A pressure correction value is determined depending on the model pressure for the current point in time and the pressure measurement value provided. The pressure correction value is assigned to the intake manifold model and at least in the first and the second operating state, the model pressure for the current state is determined depending on the pressure correction value by means of the intake manifold model.

The pressure correction value is determined, for example, in such a way that the difference between model pressure and pressure measurement value is minimized. For example, a model size of the intake manifold model that is representative of the effective cross-sectional area of the throttle valve is corrected by means of the pressure correction value such that the difference between the model pressure and the pressure measurement value is minimized. In this way, the cylinder air mass can be determined particularly precisely.

According to a further advantageous embodiment, a pressure measurement value of the gas is provided in the second operating state, which is representative of a pressure of the gas at the current time and the model pressure is adapted for the current time depending on the pressure measurement value provided.

Since the pressure of the gas essentially does not change in the second operating state, the pressure measurement value of the gas that is representative of a pressure of the gas at the current time is, for example, the pressure measurement value of the gas that is assigned to the current time or a pressure measurement value of the gas associated with the previous point in time.

In the second operating state, the values of the pressure sensor essentially do not change. Thus, in the second operating state, the model pressure can easily be adapted to the pressure measurement value. This adaptation can in turn be used when changing to the first operating state, since in the first operating state the model pressure is determined as a function of a model pressure that was determined for a previous point in time. As a result, the cylinder air mass can be determined particularly precisely and nevertheless very quickly in both operating states.

According to a further advantageous embodiment, the model pressure for the current time is adjusted depending on the pressure measurement value provided, in that the model pressure is corrected by a predetermined factor in the direction of the pressure measurement value.

In this way, the correction of the cylinder air mass can be carried out in a particularly robust and very simple manner, since, for example, very few calculation steps are necessary for the correction.

According to a further advantageous embodiment, the model pressure for the current time is adapted depending on the pressure measurement value provided, in that the model pressure is corrected in the direction of the pressure measurement value depending on the amount of the difference in the model pressure and the pressure measurement value provided.

In this way, the correction of the cylinder air mass can be carried out particularly robustly and very precisely, since the difference is used in a simple manner for the correction.

• 1 eine Brennkraftmaschine mit einer zugeordneten Steuervorrichtung,
• 2 einen Ausschnitt eines Ansaugtrakts der Brennkraftmaschine und
• 3 eine auf eine Funktion x(t) angewandte Trapezintegrationsformel

Exemplary embodiments of the invention are explained in more detail below with reference to the schematic drawings. Show it:

• 1 an internal combustion engine with an associated control device,
• 2 a section of an intake tract of the internal combustion engine and
• 3 a trapezoidal integration formula applied to a function x (t)

Elements of the same construction or function are identified with the same reference symbols in all figures.

An internal combustion engine includes an intake tract 1 , an engine block 2 , a cylinder head 3 and an exhaust tract 4 ,

The intake tract 1 preferably comprises a throttle valve 5 , a collector 6 and a suction pipe 7 that towards a cylinder Z1 via an inlet duct into a combustion chamber 9 of the engine block 2 is led. The engine block 2 includes a crankshaft 8th which have a connecting rod 10 with a piston 11 of the cylinder Z1 is coupled. The internal combustion engine includes next to the cylinder Z1 preferably other cylinders Z2 . Z3 . Z4 , However, the internal combustion engine can also comprise any other number of cylinders. The internal combustion engine is preferably arranged in a motor vehicle.

In the cylinder head 3 are preferably an injection valve 18 and a spark plug 19 arranged. Alternatively, the injection valve 18 also in the intake manifold 7 be arranged.

In the exhaust system 4 is preferably an exhaust gas catalytic converter 21 arranged, which is preferably designed as a three-way catalyst.

Furthermore, a phase adjustment device can also be provided, for example with the crankshaft 8th and an intake camshaft is coupled. The intake camshaft is with a gas intake valve 12 of the respective cylinder coupled. The phase adjustment device is designed to adjust a phase of the intake camshaft to the crankshaft 8th to enable. Furthermore, in principle, the phase adjustment device can alternatively or additionally also be designed to form a phase of an exhaust camshaft to the crankshaft 8th to adjust, the exhaust camshaft with a gas exhaust valve 13 is coupled.

Furthermore, a switching flap or another switching mechanism for changing an effective intake pipe length in the intake tract can also be used 1 be provided. In addition, one or more swirl flaps can also be provided, for example.

Furthermore, a charger can also be provided, which can be designed, for example, as an exhaust gas turbocharger and thus comprises a turbine and a compressor.

A control device 25 It is provided which sensors are assigned, which record different measured variables and each determine the measured value of the measured variable. Operating variables of the internal combustion engine include the measured variables and variables derived from the measured variables. The control device 25 is designed to determine manipulated variables as a function of at least one measured variable, which are then converted into one or more actuating signals for controlling the actuators by means of appropriate actuators. The control device 25 can also be referred to as a device for operating the internal combustion engine. The sensors are, for example, a pedal position transmitter 26 which is an accelerator pedal position of an accelerator pedal 27 detected, an air mass sensor 28 which is an air mass flow upstream of the throttle valve 5 detected, a throttle position sensor 30 that has an opening degree of the throttle valve 5 detected, an ambient pressure sensor 32 , which detects an ambient pressure of an environment of the internal combustion engine, an intake manifold pressure sensor 34 that has an intake manifold pressure in the collector 6 detected, a crankshaft angle sensor 36 , which detects a crankshaft angle, to which a speed of the internal combustion engine is then assigned. There is also an exhaust gas probe 42 provided the upstream of the catalytic converter 21 is arranged and detects, for example, a residual oxygen content of the exhaust gas of the internal combustion engine and whose measurement signal is representative of an air / fuel ratio upstream of the exhaust gas probe 42 before burning. An intake camshaft sensor or an exhaust camshaft sensor can be provided to detect the position of the intake camshaft and / or the exhaust camshaft. In addition, a temperature sensor is preferably provided, which detects an ambient temperature of the internal combustion engine, and / or a further temperature sensor is provided, the measurement signal of which is representative of one Intake air temperature in the intake tract 1 , which can also be called intake manifold temperature. Furthermore, an exhaust gas pressure sensor can also be provided, the measurement signal of which is representative of an exhaust manifold pressure, that is to say a pressure in the exhaust gas tract 4 ,

Depending on the embodiment, any subset of the sensors mentioned can be present or additional sensors can also be present.

The actuators are, for example, the throttle valve 5 who have favourited Gas Inlet and Outlet Valves 12 . 13 , the injector 18 or the phase adjuster or the spark plug 19 or an exhaust gas recirculation valve.

The air-fuel ratio, ie the ratio of participating in the combustion in the cylinder air mass m _{air, cyl,} which may also be referred to as a cylinder air mass, for participating in the combustion in the cylinder fuel mass m _fuel is an important factor influencing the pollutant emissions of an internal combustion engine , The cylinder _{air mass} m _{air, cyl} is estimated in the control _device (engine control unit) on the basis of the many available sizes and serves as the basis for the fuel metering. In order to comply with current and future pollutant emission limit values, the cylinder air mass in the engine control unit must be known to within a few percent under all stationary and transient engine operating conditions.

Pressure and temperature of the in the intake tract 1 Gases present (intake manifold pressure p _im and intake manifold temperature T _im ) are important influencing factors on the cylinder _{air mass} m _{air, cyl drawn in} by the engine and must be known as precisely as possible in order to correctly estimate the cylinder _{air mass} in the engine control unit.

The intake manifold pressure p _im can also be used as a model pressure of a gas in the intake tract 1 be designated. The intake manifold temperature T _im can also be used as a model temperature of a gas in the intake tract 1 be designated.

Modern internal combustion engines are practically always equipped with the additional temperature sensor for measuring the gas temperature in the intake tract 1 equipped, which can also be called an intake manifold temperature sensor. Typical intake manifold temperature sensors for series production show a strong PT1 behavior with time constants in the order of 5 s. In addition, modern internal combustion engines are almost always equipped with the intake manifold pressure sensor 34 and / or the air mass sensor 28 equipped with a negligible time constant (a few milliseconds). Either the measured intake manifold pressure p _{im, mes} can be used directly as a model input to determine the cylinder air mass, or a status observer (generally referred to as an intake manifold model) can be used to model the intake manifold pressure p _{im, mes} or the measured air mass flow ṁ _{air, mes} p _{im, mdl can be used} as a model _input to determine the cylinder _{air mass} . Furthermore, the intake manifold temperature can be used as a model input for determining the cylinder air mass. Either the measured intake manifold temperature T _{im, mes} direct or a corrected intake manifold _temperature T _{im, mdl is used} , for which the measured value is expanded by corrections to describe stationary warming-up _effects between the temperature sensor and the inlet valve.

As a result, all measured / observed changes in the intake manifold pressure quickly - i.e. with a delay of a few milliseconds - go into the modeling of the cylinder air mass, but changes in the intake manifold temperature only slowly with the dynamics specified by the sensor with a time constant of several seconds.

The following explains how to help to model the changes in the intake manifold pressure p _im and the intake manifold temperature T _im resulting from changing actuator positions of the internal combustion engine precisely and quickly, ie without the delay resulting from the large time constant of the temperature sensor. In particular, an intake manifold temperature modeled in this way is available more quickly than a measured value recorded with temperature sensors available for series internal combustion engines. This improves the modeling of the cylinder _{air mass} m _{air, cyl} and thus contributes to the reduction of pollutant emissions from internal combustion engines.

System limits and requirements

The system under consideration comprises the intake tract 1 an internal combustion engine with the gas therein. It is delimited by the intake manifold wall, the gas inlet valves 13 the cylinder Z1 to Z4 the internal combustion engine, the throttle valve 5 and the inlets of any further gas mass flows such as for Tank ventilation, crankcase ventilation or fuel injection. The modeling follows a 0D view, locations in the intake tract 1 are not distinguished.

In the intake tract 1 with the constant volume V _im there is a gas mass m _im with the current intake manifold pressure p _im and the current intake manifold temperature T _im ( 2 ). The general gas equation applies $$p_{im}\cdot V_{im}=m_{im}\cdot R\cdot T_{im}$$

Mass flows taken into account

mit: ṁ_in,i - Massenstrom, T_0,i - Temperatur vor Drosselstelle, p_0,i - Druck vor Drosselstelle des über die i-te Drosselstelle zufließenden Gases, $$C\left(T_{\mathrm{0,}i}\right)=\sqrt{\frac{2\cdot \kappa }{\left(\kappa -1\right)\cdot R\cdot T_{\mathrm{0,}i}}}$$

Temperaturfaktor mit κ - Isentropenexponent, R=c_p -c_v- spezifischer Gaskonstante, C_p- spezifische Wärmekapazität bei konstantem Druck, C_v- spezifische Wärmekapazität bei konstantem Volumen des zufließenden Gases, $${\mathrm{\Pi }}_i=\frac{p_{im}}{p_{\mathrm{0,}i}}$$

Druckverhältnis an der i-ten Drosselstelle, $$\mathrm{\Psi }\left({\mathrm{\Pi }}_i\right)=\{\begin{array}{cc}{\left(\frac{2}{\kappa +1}\right)}^{\frac{1}{\kappa -1}}\sqrt{\frac{\kappa -1}{\kappa +1}}& \text{für}{\mathrm{\Pi }}_{\text{i}}<\mathrm{0,53,}\text{d}.\text{h}\text{.überkritisches Druckverhältnis}\\ \sqrt{{\left(\frac{p_{im}}{p_{\mathrm{0,1}}}\right)}^{\frac{2}{\kappa }}-{\left(\frac{p_{im}}{p_{\mathrm{0,}i}}\right)}^{\frac{\kappa +1}{\kappa }}}& \text{für}{\mathrm{\Pi }}_{\text{i}}\ge \mathrm{0,53,}\text{d}.\text{h}\text{.unterkritisches Druckverhältnis}\end{array}$$

Durchflusskoeffizient an der i-ten Drosselstelle, kann für den Betriebspunkt Π_i linearisiert werden zu $$\mathrm{\Psi }\left({\mathrm{\Pi }}_i\right)={\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)-{\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot {\mathrm{\Pi }}_i$$

In general, there are several mass inflows ṁ _{in, 1} , m _{̇in, 2} , ... ṁ _{in, q} influenced by the intake manifold pressure from q sources with known gas states (ie source pressures p _0.1 , p _0.2 , ... p _{0, q} and source temperatures T _0.1 , T _0.2 , ... T _{0, q} ). These q mass inflows flow via q throttling points with the effective cross-sectional areas A _{in, 1} , A _{in, 2} , ... A _{in, q} into the intake tract 1 : $${\dot{m}}_{in.i}=A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot \mathrm{\Psi }\left({\mathrm{\Pi }}_i\right);i\in \left[1...q\right]$$

with: ṁ _{in, i} - mass flow, T _{0, i} - temperature before the throttle point, p _{0, i} - pressure before the throttle point of the gas flowing in via the i th throttle point, $$C\left(T_{0i}\right)=\sqrt{\frac{2\cdot \kappa }{\left(\kappa -1\right)\cdot R\cdot T_{0i}}}$$

Temperature factor with κ - isentropic exponent, R = c _p -c _v - specific gas constant, C _p - specific heat capacity at constant pressure, C _v - specific heat capacity at constant volume of the inflowing gas, $${\mathrm{\Pi }}_i=\frac{p_{im}}{p_{0i}}$$

Pressure ratio at the i-th throttle point, $$\mathrm{\Psi }\left({\mathrm{\Pi }}_i\right)=\{\begin{array}{cc}{\left(\frac{2}{\kappa +1}\right)}^{\frac{1}{\kappa -1}}\sqrt{\frac{\kappa -1}{\kappa +1}}& \text{For}{\mathrm{\Pi }}_{\text{i}}<0.53\text{d},\text{H}\text{.critical pressure ratio}\\ \sqrt{{\left(\frac{p_{im}}{p_{0.1}}\right)}^{\frac{2}{\kappa }}-{\left(\frac{p_{im}}{p_{0i}}\right)}^{\frac{\kappa +1}{\kappa }}}& \text{For}{\mathrm{\Pi }}_{\text{i}}\ge 0.53\text{d},\text{H}\text{. subcritical pressure ratio}\end{array}$$

Flow coefficient at the i-th throttle point can be linearized for the operating point Π _i $$\mathrm{\Psi }\left({\mathrm{\Pi }}_i\right)={\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)-{\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot {\mathrm{\Pi }}_i$$

As a simplification for everyone in the intake system 1 flowing gases assumed uniform values for isentropic exponent, gas constant and heat capacities.

• • weil das Druckverhältnis Π_i über die jeweilige Drosselstelle zumindest in manchen Betriebszuständen unterkritisch, d.h. Π_i ≥ 0,53 sein kann,
• • weil der Durchflusskoeffizient Ψ(Π_i) dann nach Gleichung ((5)) vom Saugrohrdruck p_im abhängig ist und
• • weil diese Massenströme als Funktion des Saugrohrdrucks - nicht nur als Wert - in das Saugrohrmodell eingehen sollen.

These mass inflows are considered to be influenced by the intake manifold pressure,

• Because the pressure ratio Π _i across the respective throttle point can be subcritical, at least in some operating states, ie Π _i ≥ 0.53,
• • because the flow coefficient Ψ (Π _i ) then according to equation (( 5 )) depends on the intake manifold pressure p _im and
• • Because these mass flows should be included in the intake manifold model as a function of the intake manifold pressure - not just as a value.

darstellen lassen. Examples of inflows into the intake tract influenced by the intake manifold pressure 1 are the mass flow of an external exhaust gas recirculation, the crankcase ventilation mass flow, the tank ventilation mass flow and the throttle valve mass flow that dominates in practically all operating states. It is essential that these flows to the intake manifold pressure p _in linearize, that is as linear functions of intake manifold pressure in the mold ${\dot{m}}_{in.i}=L_{in.i}*p_{im}+K_{in.i}miti\in \lfloor 1...q\rfloor$

show.

In general, there are several sinks that are influenced by the intake manifold pressure p _im in different sinks. Examples of drains from the intake tract 1 are leakage mass flows in the charged mode as well as the inlet valve mass flow dominating in practically all operating states. In practice, there is only one mass flow from the intake tract in the fault-free internal combustion engine 1 , that is the intake valve mass flow in the respective intake cylinder. This is referred to below as the outflow mass flow ṁ _out . It is in the respective engine operating point after the intake manifold pressure p _im linearized, ie as a linear function of the intake manifold pressure p _im approximated with the parameters η _slope , η _offset (slope and offset of the volumetric efficiency): $${\dot{m}}_{Out}={\eta }_{slOpe}\cdot p_{im}-{\eta }_{Offset}$$

The negative sign of the offset is not mandatory.

• - weil entweder das Druckverhältnis über die jeweilige Drosselstelle in allen Betriebszuständen überkritisch ist, d.h. Π_i < 0,53 , der Durchflusskoeffizient Ψ dann nach Gleichung ((5)) konstant ist und der jeweilige Wert des Zuflussmassenstroms unabhängig vom Saugrohrdruck p_im berechnet werden kann (z.B. am Gaseinblaseventil für CNG) oder
• - weil trotz eines möglicherweise unterkritischer Druckverhältnisses Π_i ≥ 0,53 an einer Drosselstelle als Modellvereinfachung der zugehörige Massenstrom auf Basis eines alten Werts des Saugrohrdrucks p_{im,n -1} außerhalb des Saugrohrmodells berechnet wird und dann nur als Wert (nicht als Funktion des Saugrohrdrucks) in das Saugrohrmodell eingeht.

There are, in general case, further, the intake manifold pressure p _{in the} non-affected mass flows ṁ _{in, q + 1,} m _{in, q + 2,} ..., M _{in, q + r} from r sources with known gas states (ie swelling pressures p is _{0, q +1} , p _{0, q + 2} , ... p _{0, q + r} and source temperatures T _{0, q + 1} , T _{0, q + 2} , ... T _{0, q +} r). The equations (( 2 )) to ((6)) accordingly. These mass inflows are considered not to be influenced by the intake manifold pressure p _im ,

• - because either the pressure ratio across the respective throttle point is supercritical in all operating conditions, ie Π _i <0.53, the flow coefficient Ψ then according to equation (( 5 )) is constant and the respective value of the inflow mass flow can be calculated independently of the intake manifold pressure p _im (eg on the gas injection valve for CNG) or
• - Because despite a possibly subcritical pressure ratio Π _i ≥ 0.53 at a throttle point as a model simplification, the associated mass flow is calculated on the basis of an old value of the intake manifold pressure p _{im, n -1} outside the intake manifold model and then only as a value (not as a function of the intake manifold pressure ) enters the intake manifold model.

In the intake tract 1 the law of mass conservation (mass balance) applies in general to s outflows and especially to an outflow. In the following, only an outflow is considered without restricting generality $${\dot{m}}_{im}={\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}{\dot{m}}_{in.i}}-{\displaystyle \underset{j=1}{\overset{s}{\Sigma }}{\dot{m}}_{Out.j}=}{\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}{\dot{m}}_{in.i}}-{\dot{m}}_{Out}$$

modeling

The enthalpy H _{im of} the gas in the intake tract 1 with constant volume V _im is equal to the sum of displacement work V _im · p _im, thermal energy W _therm , potential energy W _pot and kinetic energy W _{kin of} the gas in the intake tract 1 : $$H_{im}=W_{tHerm}+W_{pOt}+W_{kin}+p_{im}\cdot V_{im}$$

mit: T_im - Temperatur, m_im - Masse des Gases im Ansaugtrakt 1.The potential energy of the gas in the intake tract 1 W _pot can be neglected because there is no significant difference in height between the intake tract entry and exit and the potential energy of gases is generally negligible due to their relatively low density. The kinetic energy of the gas in the intake manifold W _kin is at least a factor in the pressure and temperature range relevant for the operation of internal combustion engines 100 smaller than the respective displacement work and thermal energy of the gas and can therefore also be neglected. As the enthalpy of the gas in the intake tract 1 arises with it $$\begin{array}{c}{H}_{im}=W_{tHerm}+p_{im}\cdot V_{im}\\ =c_v\cdot T_{im}\cdot m_{im}+p_{im}\cdot V_{im}\end{array}$$

with: T _im - temperature, m _im - mass of the gas in the intake tract 1 ,

mit: h_in,i - spezifische Enthalpie, v_in,i - Fließgeschwindigkeit, z_in,i - Höhe des i-ten Massenzuflusses, h_out- spezifische Enthalpie, v_out - Fließgeschwindigkeit, z_out- Höhe des Massenabflusses, g- Erdbeschleunigung.For the intake tract 1 The enthalpy balance is an open system with q + r inflows and an outflow, neglecting heat transfer through the intake manifold wall (which will be discussed again below) $$\frac{d}{dt}{H}_{im}={\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\cdot \left(H_{in.i}+\frac{1}{2}\cdot v_{in.i}^2+G\cdot z_{in.i}\right)\right)-{\dot{m}}_{Out}\cdot \left(H_{Out}+\frac{1}{2}\cdot v_{\mathrm{<figure-callout\; id="2"\; label="O\; u\; t"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">O\; </figure-callout>}ut}^2+G\cdot z_{Out}\right)}$$

with: h _{in, i} - specific enthalpy, v _{in, i} - flow velocity, z _{in, i} - height of the i-th mass inflow, h _out - specific enthalpy, v _out - flow velocity, z _out - height of the mass outflow, g - gravitational acceleration ,

By neglecting the kinetic and potential energy of the gas in the intake tract as described above 1 flow velocities and heights are neglected and equation ((11)) is simplified $$\frac{d}{dt}{H}_{im}={\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\cdot H_{in.i}\right)}-{\dot{m}}_{Out}\cdot H_{Out}$$

The draining masses have intake manifold temperature T _im , that is the specific enthalpy of the discharge mass flow $$H_{Out}=c_p\cdot T_{im}$$

The inflowing masses each have the temperature of their source T _{0, i} , which is the specific enthalpy of the i-th inflow mass flow $$H_{in.i}=c_p\cdot T_{0i}$$

By inserting the equations (( 10 )), ((13)), ((14)) in equation (( 12 )) surrendered $$c_v\cdot {\dot{m}}_{im}\cdot T_{im}+c_v\cdot m_{im}\cdot {\dot{T}}_{im}+{\dot{p}}_{im}\cdot V_{im}+p_{im}\cdot {\dot{V}}_{im}=c_p\cdot {\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\cdot T_{0i}\right)-{\dot{m}}_{Out}\cdot c_p\cdot T_{im}}$$

eine erste implizite Differentialgleichung erster Ordnung der Variablen Saugrohrdruck p_im und Saugrohrtemperatur T_im .Because of the constant intake manifold volume, p _{im ·} V̇ _im = 0. Taking into account the functional dependencies from ((1)), ((2)), ((4)) and ((8)), ((15)) results from rearrangement $${\dot{p}}_{im}\cdot V_{im}=c_p\cdot \left({\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot T_{0i}\right)-{\dot{m}}_{Out}\left(p_{im}\right)\cdot T_{im}}\right)-c_v\cdot \left({\dot{m}}_{im}\left(p_{im}\right)\cdot T_{im}+m_{im}\left(p_{im}.T_{im}\right)\cdot {\dot{T}}_{im}\right)$$

a first implicit first order differential equation of the intake manifold pressure variables p _im and intake manifold temperature T _im ,

Derivation of the general gas equation for the gas in the intake tract 1 ((1)) after time returns $${\dot{p}}_{im}\cdot V_{im}+p_{im}\cdot {\dot{V}}_{im}={\dot{m}}_{im}\cdot R\cdot T_{im}+m_{im}\cdot R\cdot {\dot{T}}_{im}$$

eine zweite implizite Differentialgleichung erster Ordnung der Variablen Saugrohrdruck p_im und Saugrohrtemperatur T_im .Because of the constant intake manifold volume, p _im · V̇ _im = 0. Taking into account the functional dependencies from ((1)), ((2)), ((4)) and ((8)) we get from ((17)) $${\dot{p}}_{im}=V_{im}={\dot{m}}_{im}\left(p_{im}\right)\cdot R\cdot T_{im}+m_{im}\left(p_{im}.T_{im}\right)\cdot R\cdot {\dot{T}}_{im}$$

a second implicit first order differential equation of the intake manifold pressure variables p _im and intake manifold temperature T _im ,

Discretization of the model

The two first order differential equations of the variable intake manifold pressure p _im and intake manifold temperature T _im ((16)) and ((18)) are reshaped so that the intake manifold pressure gradient ṗ _im and the intake manifold temperature gradient Ṫ _{im are} eliminated.

The difference of the equations (( 18 )) - ((16)) eliminates the intake manifold pressure gradient ṗ _im . After inserting the mass balance (( 8th )) provides the change to Ṫ _in $${\dot{T}}_{im}=\frac{1}{m_{im}\left(p_{im}.T_{im}\right)}\cdot {\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot \left(T_{0i}-T_{im}\right)\right)}$$

mit der Abtastzeit (sampling time) t_s =t_n -t_n-1 angewandt: $$T_{im}=T_{im,n}=T_{im,n-1}+\frac{t_s}{2}\cdot \left({\dot{T}}_{im,n-1}+{\dot{T}}_{im}\right)$$

On the intake manifold temperature T _im the general trapezoid integration formula (see 3 ) to discretize the model over time $$x_n=x\left(t_n\right)=x\left(t_{n-1}\right)+\frac{\dot{x}\left(t_{n-1}\right)+\dot{x}\left(t_n\right)}{2}\cdot \left(t_n-t_{n-1}\right)$$

with the sampling time t _s = t _n -t _n-1 applied: $$T_{im}=T_{im.n}=T_{im.n-1}+\frac{t_s}{2}\cdot \left({\dot{T}}_{im.n-1}+{\dot{T}}_{im}\right)$$

Old intake manifold temperature T _{im, n-1} and old intake manifold temperature gradient T _{im, n-1} are known values at time n from the previous calculation step n-1. By inserting equation ((19)) in ((21)), the intake manifold temperature gradient Ṫ _im is also eliminated: $$T_{im}=T_{im.n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im.n-1}+\frac{t_s}{2\cdot m_{im}\left(p_{im}.T_{im}\right)}\cdot {\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot \left(T_{0i}-T_{im}\right)\right)}$$

The terms with values known at the beginning of the calculations for time n are summarized to simplify the further derivation: $$G_{0.1}=T_{im.n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im.n-1}$$

würde die in T_im lineare Gleichung ((22)) mit einem quadratischen Term T_im ² verkomplizieren. Da die Gasmasse im Ansaugtrakt 1 nicht springen kann und sich in einem Rechenschritt nur relativ wenig verändert, kann ohne großen Genauigkeitsverlust zur Vereinfachung von Gleichung ((22)) die aktuelle, unbekannte Gasmasse m_im durch die alte, im vorhergehenden Rechenschritt bestimmte Gasmasse $m_{im,n-1}=\frac{V_{im}}{R}\cdot \frac{p_{im,n-1}}{T_{im,n-1}}$

ersetzt werden: $$T_{im}=G_{\mathrm{0,1}}+\frac{t_s\cdot R\cdot T_{im,n-1}}{2\cdot V_{im}\cdot p_{im,n-1}}\cdot {\displaystyle \sum _{i=1}^{q+r}\left({\dot{m}}_{in,i}\left(p_{im}\right)\cdot \left(T_{\mathrm{0,}i}-T_{im}\right)\right)}$$

A replacement of the current gas mass in the intake tract 1 in equation (( 22 )) according to equation (( 1 )) $$m_{im}=\frac{p_{im}\cdot V_{im}}{R\cdot T_{im}}$$

would be the linear equation in T _in (( 22 )) with a quadratic term T _in ² . Because the gas mass in the intake tract 1 cannot jump and changes only relatively little in one calculation step, the current, unknown gas mass m _{in the} gas mass determined by the old calculation step in the previous calculation step can be simplified without great loss of accuracy to simplify equation ((22)) $m_{im.n-1}=\frac{V_{im}}{R}\cdot \frac{p_{im.n-1}}{T_{im.n-1}}$

be replaced: $$T_{im}=G_{0.1}+\frac{t_s\cdot R\cdot T_{im.n-1}}{2\cdot V_{im}\cdot p_{im.n-1}}\cdot {\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot \left(T_{0i}-T_{im}\right)\right)}$$

The terms with values known at the start of the calculations for time n are summarized in order to simplify the further derivation $$e=\frac{t_s\cdot R\cdot T_{im.n-1}}{2\cdot V_{im}\cdot p_{im.n-1}}$$

The inflows influenced by the intake manifold pressure and the r inflows unaffected by the intake manifold pressure are written separately $$\begin{array}{c}{T}_{im}=G_{0.1}+e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot \left(T_{0i}-T_{im}\right)\right)+e\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot \left(T_{0k}-T_{im}\right)\right)}}\\ =G_{0.1}+e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot \left(T_{0i}-T_{im}\right)\right)+e\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}{(\dot{m}}_{in.k}\cdot T{}_{0k})-e\cdot T_{im}\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}(\dot{m}{}_{in.k})}}}\end{array}$$

und $$G_{\mathrm{1,1}}=-E\cdot {\displaystyle \sum _{k=1}^r\left({\dot{m}}_{in,k}\right)}$$

The terms with values known at the start of the calculations for time n are summarized in order to simplify the further derivation $$G_{0.2}=e\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)}$$

and $$G_{1.1}=-e\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\right)}$$

Replacing the inflows influenced by the intake manifold pressure according to equation ((2)) in ((27)) yields $$T_{im}=G_{0.1}+e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot \mathrm{\Psi }\left({\mathrm{\Pi }}_i\right)\cdot \left(T_{0i}-T_{im}\right)\right)}+G_{0.2}+G_{1.1}\cdot T_{im}$$

$$\begin{array}{c}{T}_{im}=G_{\mathrm{0,1}}+G_{\mathrm{0,2}}+G_{\mathrm{1,1}}\cdot T_{im}\\ +E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\cdot \left(T_{\mathrm{0,}i}-T_{im}\right)\right)}\\ -p_{im}\cdot E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot \left(T_{\mathrm{0,}i}-T_{im}\right)\right)}\end{array}$$

und $$\begin{array}{c}{T}_{im}=G_{\mathrm{0,1}}+G_{\mathrm{0,2}}+G_{\mathrm{1,1}}\cdot T_{im}\\ +E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}\\ -T_{im}\cdot E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\right)}\\ -p_{im}\cdot E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}\\ +p_{im}\cdot T_{im}\cdot E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\right)}\end{array}$$

Replacing the flow coefficient at the i-th throttle point according to equations ((4)) and ((6)) in ((30)) yields $$\begin{array}{c}{T}_{im}=G_{0.1}+G_{0.2}+G_{1.1}\cdot T_{im}\\ +e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot \left({\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)-{\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot \frac{p_{im}}{p_{0i}}\right)\cdot \left(T_{0i}-T_{im}\right)\right)}\end{array}$$

$$\begin{array}{c}{T}_{im}=G_{0.1}+G_{0.2}+G_{1.1}\cdot T_{im}\\ +e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\cdot \left(T_{0i}-T_{im}\right)\right)}\\ -p_{im}\cdot e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot \left(T_{0i}-T_{im}\right)\right)}\end{array}$$

and $$\begin{array}{c}{T}_{im}=G_{0.1}+G_{0.2}+G_{1.1}\cdot T_{im}\\ +e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}\\ -T_{im}\cdot e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\right)}\\ -p_{im}\cdot e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}\\ +p_{im}\cdot T_{im}\cdot e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\right)}\end{array}$$

$$G_{\mathrm{1,2}}=-E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\right)}$$

$$G_{\mathrm{2,1}}=-E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}$$

und $$G_{\mathrm{3,1}}=E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\right)}$$

The terms with values known at the start of the calculations for time n are summarized in order to simplify the further derivation $$G_{0.3}=e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}$$

$$G_{1.2}=-e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\right)}$$

$$G_{2.1}=-e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}$$

and $$G_{3.1}=e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\right)}$$

und weiter zu $$0=\frac{G_{\mathrm{0,1}}+G_{\mathrm{0,2}}+G_{\mathrm{0,3}}}{G_{\mathrm{3,1}}}+\frac{G_{\mathrm{1,1}}+G_{\mathrm{1,2}}-1}{G_{\mathrm{3,1}}}\cdot T_{im}+\frac{G_{\mathrm{2,1}}}{G_{\mathrm{3,1}}}\cdot p_{im}+p_{im}\cdot T_{im}$$

Equation (( 33 )) simplifies itself $$T_{im}=\left(G_{0.1}+G_{0.2}+G_{0.3}\right)+\left(G_{1.1}+G_{1.2}\right)\cdot T_{im}+G_{2.1}\cdot p_{im}+G_{3.1}\cdot p_{im}\cdot T_{im}$$

and on to $$0=\frac{G_{0.1}+G_{0.2}+G_{0.3}}{G_{3.1}}+\frac{G_{1.1}+G_{1.2}-1}{G_{3.1}}\cdot T_{im}+\frac{G_{2.1}}{G_{3.1}}\cdot p_{im}+p_{im}\cdot T_{im}$$

Analogous to eliminating the intake manifold pressure gradient in equation (( 19 )) ff. In a second parallel transformation, the intake manifold temperature gradient is derived from the system of equations (( 16 )), ((18)) eliminated. A multiplication of equation (( 16 )) with the specific gas constant R. $${\dot{p}}_{im}\cdot V_{im}=c_p\cdot \left({\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}{(\dot{m}}_{in.i}}\left(p_{im}\right)\cdot T{}_{0i})-{\dot{m}}_{Out}\left(p_{im}\right)\cdot T_{im}\right)-c_v\cdot \left({\dot{m}}_{im}\left(p_{im}\right)\cdot T_{im}+m_{im}\left(p_{im}.T_{im}\right)\cdot {\dot{T}}_{im}\right)$$

A multiplication of equation (( 18 )) with the specific heat capacity c _v $$c_v\cdot {\dot{p}}_{im}\cdot V_{im}=c_v\cdot {\dot{m}}_{im}\left(p_{im}\right)\cdot R\cdot T_{im}+c_v\cdot m_{im}\left(p_{im}.T_{im}\right)\cdot R\cdot {\dot{T}}_{im}$$

und unter Berücksichtigung der Definition der spezifischen Gaskonstante R=c_p -c_v $${\dot{p}}_{im}=\frac{R}{V_{im}}\cdot \left({\displaystyle \sum _{i=1}^{q+r}\left({\dot{m}}_{in,i}\left(p_{im}\right)\cdot T_{\mathrm{0,}i}\right)-{\dot{m}}_{out}}\left(p_{in}\right)\cdot T_{im}\right)$$

The sum of the equations (( 40 )) and ((41)) results $$\left(c_v+R\right)\cdot {\dot{p}}_{im}\cdot V_{im}=R\cdot c_p\cdot \left({\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{in}\right)\cdot T_{0i}\right)-{\dot{m}}_{Out}}\left(p_{in}\right)\cdot T_{im}\right)$$

and taking into account the definition of the specific gas constant R = c _p -c _v $${\dot{p}}_{im}=\frac{R}{V_{im}}\cdot \left({\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot T_{0i}\right)-{\dot{m}}_{Out}}\left(p_{in}\right)\cdot T_{im}\right)$$

On the intake manifold pressure _P is the general trapezoidal integration formula (( 20 )) with the sampling time t _s = t _n -t _n-1 applied: $$p_{im}=p_{im.n}=p_{im.n-1}+\frac{t_s}{2}\cdot \left({\dot{p}}_{im.n-1}+{\dot{p}}_{im}\right)$$

Old intake manifold pressure P _{im, n-1} and old intake manifold pressure gradient ṗ _{im, n-1} are known values at time n from the previous calculation step n-1. By inserting equation ((43)) into ((44)), the intake manifold pressure gradient ṗ _im is also eliminated $$p_{im}=p_{im.n-1}+\frac{t_s}{2}\cdot {\dot{p}}_{im.n-1}+\frac{t_s}{2}\cdot \frac{R}{V_{im}}\cdot \left({\displaystyle \underset{i=1}{\overset{q+r}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot T_{0i}\right)-{\dot{m}}_{Out}\left(p_{im}\right)\cdot T_{im}}\right)$$

und $$D=\frac{t_s\cdot R}{2\cdot V_{im}}$$

The terms with values known at the beginning of the calculations for time n are summarized to simplify the further derivation: $$F_{0.1}=p_{im.n-1}{+}_2^{t_s}\cdot {\dot{p}}_{im.n-1}$$

and $$D=\frac{t_s\cdot \mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">R</figure-callout>}}{2\cdot V_{im}}$$

The inflows influenced by the intake manifold pressure and the r inflows unaffected by the intake manifold pressure are written separately $$p_{im}=F_{0.1}+D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left({\dot{m}}_{in.i}\left(p_{im}\right)\cdot T_{0i}\right)+D\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)-D\cdot {\dot{m}}_{Out}\left(p_{im}\right)\cdot T_{im}}}$$

Replacing the discharge mass flow according to equation (( 7 )) and the inflow mass flows according to equation (( 2 )) results $$p_{im}=F_{0.1}+D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot \mathrm{\Psi }\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)+D\cdot }{\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)-D\cdot {\dot{m}}_{Out}\left(p_{im}\right)\cdot T_{im}}$$

und $$\begin{array}{cc}{p}_{im}& =F_{\mathrm{0,1}}\\ & +D\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}\\ & -D\cdot p_{im}\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}\\ & +D\cdot {\displaystyle \sum _{k=1}^r\left({\dot{m}}_{in,k}\cdot T_{\mathrm{0,}k}\right)}\\ & +D\cdot {\eta }_{offset}\cdot T_{im}\\ & -D\cdot {\eta }_{slope}\cdot p_{im}\cdot T_{im}\end{array}$$

Replacing the flow coefficient at the i-th throttle point according to equations ((4)) and ((6)) in ((49)) yields $$\begin{array}{c}{p}_{im}=F_{0.1}\\ +D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot \left({\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)-{\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot \frac{p_{im}}{p_{0i}}\right)\cdot T_{0i}\right)}\\ +D\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)}\\ -D\cdot \left({\eta }_{slOpe}\cdot p_{im}-{\eta }_{Offset}\right)\cdot T_{im}\end{array}$$

and $$\begin{array}{cc}{p}_{im}& =F_{0.1}\\ & +D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}\\ & -D\cdot p_{im}\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}\\ & +D\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)}\\ & +D\cdot {\eta }_{Offset}\cdot T_{im}\\ & -D\cdot {\eta }_{slOpe}\cdot p_{im}\cdot T_{im}\end{array}$$

$$F_{\mathrm{0,3}}=D\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}$$

$$F_{\mathrm{1,1}}=D\cdot {\eta }_{offset}$$

$$F_{\mathrm{2,1}}=-D\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}$$

und $$F_{\mathrm{3,1}}=-D\cdot {\eta }_{slope}$$

The terms with values known at the beginning of the calculations for time n are summarized to simplify the further derivation: $$F_{0.2}=D\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)}$$

$$F_{0.3}=D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}$$

$$F_{1.1}=D\cdot {\eta }_{Offset}$$

$$F_{2.1}=-D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}$$

and $$F_{3.1}=-D\cdot {\eta }_{slOpe}$$

und weiter zu $$0=\frac{F_{\mathrm{0,1}}+F_{\mathrm{0,2}}+F_{\mathrm{0,3}}}{F_{\mathrm{3,1}}}+\frac{F_{\mathrm{1,1}}}{F_{\mathrm{3,1}}}\cdot T_{im}+\frac{F_{\mathrm{2,1}}-1}{F_{\mathrm{3,1}}}\cdot p_{im}+p_{im}\cdot T_{im}$$

Equation (( 51 )) simplifies itself $$p_{im}=\left(F_{0.1}+F_{0.2}+F_{0.3}\right)+F_{1.1}\cdot T_{im}+F_{2.1}\cdot p_{im}+F_{3.1}\cdot p_{im}\cdot T_{im}$$

and on to $$0=\frac{F_{0.1}+F_{0.2}+F_{0.3}}{F_{3.1}}+\frac{F_{1.1}}{F_{3.1}}\cdot T_{im}+\frac{F_{2.1}-1}{F_{3.1}}\cdot p_{im}+p_{im}\cdot T_{im}$$

Solution of the system of equations

und $$p_{im}\cdot T_{im}+d\cdot p_{im}+e\cdot T_{im}+f=0$$

mit $$\begin{array}{ll}a=\frac{G_{\mathrm{2,1}}}{G_{\mathrm{3,1}}},\hfill & d=\frac{F_{\mathrm{2,1}}-1}{F_{\mathrm{3,1}}},\hfill \\ b=\frac{G_{\mathrm{1,1}}+G_{\mathrm{1,2}}-1}{G_{\mathrm{3,1}}},\hfill & e=\frac{F_{\mathrm{1,1}}}{F_{\mathrm{3,1}}},\hfill \\ c=\frac{G_{\mathrm{0,1}}+G_{\mathrm{0,2}}+G_{\mathrm{0,3}}}{G_{\mathrm{3,1}}},\hfill & f=\frac{F_{\mathrm{0,1}}+F_{\mathrm{0,2}}+F_{\mathrm{0,3}}}{F_{\mathrm{3,1}}}.\hfill \end{array}$$

The equations (( 39 )) and ((58)) form a system of equations of the variables intake manifold pressure p _im and intake manifold temperature T _{im in} the form $$p_{im}\cdot T_{im}+a\cdot p_{im}+b\cdot T_{im}+c=0$$

and $$p_{im}\cdot T_{im}+d\cdot p_{im}+e\cdot T_{im}+f=0$$

With $$\begin{array}{ll}a=\frac{G_{2.1}}{G_{3.1}}.\hfill & d=\frac{F_{2.1}-1}{F_{3.1}}.\hfill \\ b=\frac{G_{1.1}+G_{1.2}-1}{G_{3.1}}.\hfill & e=\frac{F_{1.1}}{F_{3.1}}.\hfill \\ c=\frac{G_{0.1}+G_{0.2}+G_{0.3}}{G_{3.1}}.\hfill & f=\frac{F_{0.1}+F_{0.2}+F_{0.3}}{F_{3.1}},\hfill \end{array}$$

The linearized intake manifold model at the current operating point results from the difference between equations ((59)) and ((60)) $$\left(a-d\right)\cdot p_{im}+\left(b-e\right)\cdot T_{im}+c-f=0$$

For b = e, according to equation ((61)), any changes in the intake manifold temperature would not cause a change in the intake manifold pressure, which contradicts the general gas equation ((1)). The case b = e is therefore not physically relevant. Equation ((61)) can be changed for b ≠ e $$T_{im}=\frac{d-a}{b-e}\cdot p_{im}+\frac{f-c}{b-e}$$

Substituting equation ((62)) into either equation ((59)) or ((60)) results in each case $$\left(d-a\right)\cdot p_{im}{}^2+\left(f-c-a\cdot e+b\cdot d\right)\cdot p_{im}+\left(b\cdot f-c\cdot e\right)=0$$

For a = d, according to equation ((61)), any changes in the intake manifold pressure would not cause a change in the intake manifold temperature, which contradicts the general gas equation ((1)).

The case a = d is therefore not physically relevant. Equation ((63)) can be changed for a ≠ d $$p_{im}{}^2+\frac{f-c-a\cdot e+b\cdot d}{d-a}\cdot p_{im}+\frac{b\cdot f-c\cdot e}{d-a}=x^2+P\cdot x+Q=0$$

liefert für die praktisch relevanten Fälle für den Zeitpunkt n immer zwei Lösungen. Als Näherung des Saugrohrdrucks für den Zeitpunkt n wird wegen der real vorhandenen Stetigkeit des Saugrohrdrucks jeweils die näher an der alten Lösung für den Zeitpunkt n-1 liegende Lösung verwendet.The solution formula of the quadratic equation $$\begin{array}{cc}{p}_{im}& =-\frac{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">P</figure-callout>}}{2}+\sqrt{\frac{{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">P</figure-callout>}}^2}{4}-Q}\\ & =-\frac{f-c-a\cdot e+b\cdot \mathrm{<figure-callout\; id="2"\; label="d"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">d</figure-callout>}}{2\cdot \left(d-a\right)}\pm \sqrt{\frac{{\left(f-c-a\cdot e+b\cdot d\right)}^2}{4\cdot {\left(d-a\right)}^2}-\frac{b\cdot f-c\cdot e}{d-a}}\end{array}$$

always provides two solutions for the practically relevant cases for time n. As an approximation of the intake manifold pressure for the point in time n, the solution closer to the old solution for the point in time n-1 is used in each case because of the actually existing continuity of the intake manifold pressure.

und $$T_{im,mdl}=\frac{d-a}{b-e}\cdot p_{im}+\frac{f-c}{b-e}$$

mit $$a=\frac{G_{\mathrm{2,1}}}{G_{\mathrm{3,1}}},$$

$$b=\frac{G_{\mathrm{1,1}}+G_{\mathrm{1,2}}-1}{G_{\mathrm{3,1}}},$$

$$c=\frac{G_{\mathrm{0,1}}+G_{\mathrm{0,2}}+G_{\mathrm{0,3}}}{G_{\mathrm{3,1}}},$$

$$d=\frac{F_{\mathrm{2,1}}-1}{F_{\mathrm{3,1}}},$$

$$e=\frac{F_{\mathrm{1,1}}}{F_{\mathrm{3,1}}},$$

$$f=\frac{F_{\mathrm{0,1}}+F_{\mathrm{0,2}}+F_{\mathrm{0,3}}}{F_{\mathrm{3,1}}},$$

$$D=\frac{t_s\cdot R}{2\cdot V_{im}},$$

$$E=\frac{t_s\cdot R\cdot T_{im,mdl,n-1}}{2\cdot V_{im}\cdot p_{im,mdl,n-1}},$$

$$F_{\mathrm{0,1}}=p_{im,mdl,n-1}+\frac{t_s}{2}\cdot {\dot{p}}_{im,mdl,n-1},$$

$$F_{\mathrm{0,2}}=D\cdot {\displaystyle \sum _{k=1}^r\left({\dot{m}}_{in,k}\cdot T_{\mathrm{0,}k}\right)},$$

$$F_{\mathrm{0,3}}=D\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)},$$

$$F_{\mathrm{1,1}}=D\cdot {\eta }_{offset},$$

$$F_{\mathrm{2,1}}=-D\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)},$$

$$F_{\mathrm{3,1}}=-D\cdot {\eta }_{slope},$$

$$G_{\mathrm{0,1}}=T_{im,mdl,n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im,mdl,n-1},$$

$$G_{\mathrm{0,2}}=E\cdot {\displaystyle \sum _{k=1}^r\left({\dot{m}}_{in,k}\cdot T_{\mathrm{0,}k}\right)},$$

$$G_{\mathrm{0,3}}=E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)},$$

$$G_{\mathrm{1,1}}=-E\cdot {\displaystyle \sum _{k=1}^r\left({\dot{m}}_{in,k}\right)},$$

$$G_{\mathrm{1,2}}=-E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot p_{\mathrm{0,}i}\cdot C\left(T_{\mathrm{0,}i}\right){\mathrm{\Psi }}_{offset}\left({\mathrm{\Pi }}_i\right)\right)},$$

$$G_{\mathrm{2,1}}=-E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\cdot T_{\mathrm{0,}i}\right)}$$

und $$G_{\mathrm{3,1}}=E\cdot {\displaystyle \sum _{i=1}^q\left(A_{in,i}\cdot C\left(T_{\mathrm{0,}i}\right)\cdot {\mathrm{\Psi }}_{slope}\left({\mathrm{\Pi }}_i\right)\right)}.$$

In summary, intake manifold pressure p _im and intake manifold temperature T _im for the time n from the equations ((60)), ((62)) and ((65)) are modeled as $$p_{im.mdl}=-\frac{f-c-a\cdot e+b\cdot \mathrm{<figure-callout\; id="2"\; label="d"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">d</figure-callout>}}{2\cdot \left(d-a\right)}\pm \sqrt{\frac{{\left(f-c-a\cdot e+b\cdot d\right)}^2}{4\cdot {\left(d-a\right)}^2}-\frac{b\cdot f-c\cdot e}{d-a}}$$

and $$T_{im.mdl}=\frac{d-a}{b-e}\cdot p_{im}+\frac{f-c}{b-e}$$

With $$a=\frac{G_{2.1}}{G_{3.1}}.$$

$$b=\frac{G_{1.1}+G_{1.2}-1}{G_{3.1}}.$$

$$c=\frac{G_{0.1}+G_{0.2}+G_{0.3}}{G_{3.1}}.$$

$$d=\frac{F_{2.1}-1}{F_{3.1}}.$$

$$e=\frac{F_{1.1}}{F_{3.1}}.$$

$$f=\frac{F_{0.1}+F_{0.2}+F_{0.3}}{F_{3.1}}.$$

$$D=\frac{t_s\cdot \mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">R</figure-callout>}}{2\cdot V_{im}}.$$

$$e=\frac{t_s\cdot R\cdot T_{im.mdl.n-1}}{2\cdot V_{im}\cdot p_{im.mdl.n-1}}.$$

$$F_{0.1}=p_{im.mdl.n-1}+\frac{t_s}{2}\cdot {\dot{p}}_{im.mdl.n-1}.$$

$$F_{0.2}=D\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)}.$$

$$F_{0.3}=D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}.$$

$$F_{1.1}=D\cdot {\eta }_{Offset}.$$

$$F_{2.1}=-D\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}.$$

$$F_{3.1}=-D\cdot {\eta }_{slOpe}.$$

$$G_{0.1}=T_{im.mdl.n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im.mdl.n-1}.$$

$$G_{0.2}=e\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\cdot T_{0k}\right)}.$$

$$G_{0.3}=e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}.$$

$$G_{1.1}=-e\cdot {\displaystyle \underset{k=1}{\overset{r}{\Sigma }}\left({\dot{m}}_{in.k}\right)}.$$

$$G_{1.2}=-e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot p_{0i}\cdot C\left(T_{0i}\right){\mathrm{\Psi }}_{Offset}\left({\mathrm{\Pi }}_i\right)\right)}.$$

$$G_{2.1}=-e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\cdot T_{0i}\right)}$$

and $$G_{3.1}=e\cdot {\displaystyle \underset{i=1}{\overset{q}{\Sigma }}\left(A_{in.i}\cdot C\left(T_{0i}\right)\cdot {\mathrm{\Psi }}_{slOpe}\left({\mathrm{\Pi }}_i\right)\right)},$$

Adjustment of the intake manifold model to the measured gas condition

• 1. Beobachterkorrektur: Beispielsweise ein oder mehrere Eingänge des Modells können automatisch so korrigiert werden, dass die Modellabweichungen T_im,mes - T_im,mdl und/oder p_im,mes - p_im,mdl minimiert werden.

In quasi-stationary operation, i.e. after all the input signals to the intake manifold model have been essentially constant for several seconds, it is advantageous if the intake manifold _{model measures the} intake manifold pressure p _{im, mdl} = p _{im, mes} and the measurable intake manifold _temperature T _{im, mdl} = T _{im, mes} outputs. The shape of the intake manifold model given by equations ((66)) and ((67)) cannot ensure this, since they are not dependent on the measured intake manifold pressure p _{im, mes} or the measured intake manifold temperature T _{im, mes} . In particular, the neglect of heat transfers through the intake manifold wall, assumed for equation ((11)), significantly falsifies the intake manifold model in a stationary manner. In order to merge measured values and model outputs in a stationary manner, three methods are possible:

• 1. Observer correction: For example, one or more inputs of the model can be corrected automatically so that the model deviations T _{im, mes} - T _{im, mdl} and / or p _{im, mes} - p _{im, mdl are} minimized.

For this purpose, a temperature measurement value of the gas is provided in the quasi-stationary operation, which is representative of a temperature of the gas at the current time. A temperature correction value is determined depending on the model temperature for the current time and the temperature measurement value provided. The temperature correction value is assigned to the intake manifold model and at least in one Unsteady-state operation and quasi-steady-state operation, the model temperature for the current state is determined depending on the temperature correction value by means of the intake manifold model.

The temperature correction value is determined, for example, in such a way that the difference between the model temperature and the temperature measured value is minimized. For example, the model variable “temperature of the throttle valve mass flow” of the intake manifold model is corrected using the temperature correction value. As an alternative or in addition, it is also possible to introduce an additional, non-physically modeled model input “heat flow through the intake manifold wall”, which is corrected using the temperature correction value in such a way that the difference between the model temperature and the temperature measured value is minimized.

Alternatively or additionally, a pressure measurement value of the gas which is representative of a pressure of the gas at the current time is provided in the quasi-stationary operation. A pressure correction value is determined depending on the model pressure for the current point in time and the pressure measurement value provided. The pressure correction value is assigned to the intake manifold model and at least in the first and the second operating state, the model pressure for the current state is determined depending on the pressure correction value by means of the intake manifold model.

The pressure correction value is determined, for example, in such a way that the difference between model pressure and pressure measurement value is minimized. For example, a model size of the intake manifold model that is representative of the effective cross-sectional area of the throttle valve is corrected by means of the pressure correction value such that the difference between the model pressure and the pressure measurement value is minimized.

und $$T_{im,mdl,cor1}=T_{im,mdl}+\text{sgn}\left(T_{im,mes}-T_{im,mdl}\right)\cdot T_{inc}$$

2. Incremental model correction: As an alternative or in addition, the model temperature and / or the model pressure for the current point in time are adjusted depending on the temperature measurement value and / or pressure measurement value provided by the model temperature and / or the model pressure in the direction of the temperature measurement value and / or of the pressure reading are corrected. For this purpose, in particular the model _outputs T _{im, mdl} and p _{im, mdl} from equations ((66)), ((67) are shifted in each scanning step by predetermined increments T _{im, inc} and p _{im, inc} to be measured in the direction of measured values: $$p_{im.mdl.\mathrm{<figure-callout\; id="1"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png,DE102014209793B4\_0103.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or1}=p_{im.mdl}+\text{sgn}\left(p_{im.mes}-p_{im.mdl}\right)\cdot T_{inc}$$

and $$T_{im.mdl.\mathrm{<figure-callout\; id="1"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png,DE102014209793B4\_0103.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or1}=T_{im.mdl}+\text{sgn}\left(T_{im.mes}-T_{im.mdl}\right)\cdot T_{inc}$$

$$F_{\mathrm{0,1}}=p_{im,mdl,cor\mathrm{1,}n-1}+\frac{t_s}{2}\cdot {\dot{p}}_{im,mdl,cor\mathrm{1,}n-1}$$

und $$G_{\mathrm{0,1}}=T_{im,mdl,cor\mathrm{1,}n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im,mdl,cor\mathrm{1,}n-1}.$$

The parameters of the intake manifold model ((66)), ((67)) must be corrected accordingly: $$e=\frac{t_s\cdot R\cdot T_{im.mdl.\mathrm{<figure-callout\; id="1"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png,DE102014209793B4\_0103.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{1,}n-1}}{2\cdot V_{im}\cdot p_{im.mdl.\mathrm{<figure-callout\; id="1"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png,DE102014209793B4\_0103.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{1,}n-1}}$$

$$F_{0.1}=p_{im.mdl.\mathrm{<figure-callout\; id="1"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png,DE102014209793B4\_0103.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{1,}n-1}+\frac{t_s}{2}\cdot {\dot{p}}_{im.mdl.\mathrm{<figure-callout\; id="1"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png,DE102014209793B4\_0103.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{1,}n-1}$$

and $$G_{0.1}=T_{im.mdl.cOr\mathrm{1,}n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im.mdl.cOr\mathrm{1,}n-1},$$

und $$T_{im,mdl,cor2}=T_{im,mdl}+\left(T_{im,mes}-T_{im,mdl}\right)\cdot F_{T_{inc}}$$

3. Proportional model correction: Alternatively or additionally, the model temperature and / or the model pressure for the current point in time are adjusted depending on the temperature measurement value and / or pressure measurement value provided, by the model temperature depending on the amount of the difference between the model temperature and the temperature measurement value provided in the direction of the temperature measurement value is corrected and / or in that the model pressure is corrected in the direction of the pressure measurement value depending on the amount of the difference in the model pressure and the pressure measurement value provided. In particular, the model _outputs T _{im, mdl} and p _{im, mdl} from equations ((66)), ((67) are shifted in each scanning step by parts of the model errors FT _{im, inc} and Fp _{im, inc} to be calibrated in the direction of measured values: $$p_{im.mdl.\mathrm{<figure-callout\; id="2"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or2}=p_{im.mdl}+\left(p_{im.mes}-p_{im.mdl}\right)\cdot F_{p_{inc}}$$

and $$T_{im.mdl.\mathrm{<figure-callout\; id="2"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or2}=T_{im.mdl}+\left(T_{im.mes}-T_{im.mdl}\right)\cdot F_{T_{inc}}$$

$$F_{\mathrm{0,1}}=p_{im,mdl,cor\mathrm{2,}n-1}+\frac{t_s}{2}\cdot {\dot{p}}_{im,mdl,cor\mathrm{2,}n-1}$$

und $$G_{\mathrm{0,1}}=T_{im,mdl,cor\mathrm{2,}n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im,mdl,cor\mathrm{2,}n-1}.$$

The parameters of the intake manifold model ((66)), ((67)) must be corrected accordingly: $$e=\frac{t_s\cdot R\cdot T_{im.mdl.\mathrm{<figure-callout\; id="2"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{2,}n-1}}{2\cdot V_{im}\cdot p_{im.mdl.\mathrm{<figure-callout\; id="2"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{2,}n-1}}$$

$$F_{0.1}=p_{im.mdl.\mathrm{<figure-callout\; id="2"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{2,}n-1}+\frac{t_s}{2}\cdot {\dot{p}}_{im.mdl.\mathrm{<figure-callout\; id="2"\; label="c\; O\; r"\; filenames="DE102014209793B4\_0102.png"\; state="\{\{state\}\}">c\; </figure-callout>}Or\mathrm{2,}n-1}$$

and $$G_{0.1}=T_{im.mdl.cOr\mathrm{2,}n-1}+\frac{t_s}{2}\cdot {\dot{T}}_{im.mdl.cOr\mathrm{2,}n-1},$$

With the proposed method, the influence of rapid changes in the temperature of the gas in the intake tract 1 on the cylinder air mass can be described more precisely in series engine control devices than is possible on the basis of a measurement with a temperature sensor available for series engines. Due to the more precise fuel metering as a result of the more precise determination of the cylinder air mass, pollutant emissions of the internal combustion engine can be reduced.

The control device 25 is designed to carry out the procedure described above and thus in particular to determine the cylinder air mass which is in the respective cylinder after the gas exchange valves have been closed.

The control device has, in particular, a program and data memory and a corresponding computing unit, such as a microprocessor.

Claims (8)

Method for operating an internal combustion engine comprising an intake tract (1) and one or more cylinders (Z1 to Z4), to which gas inlet valves (12) and gas outlet valves (13) are assigned, wherein gas exchange valves include gas inlet valves (12) and gas outlet valves (13) which - in a first operating state - a model temperature of a gas in the intake tract (1) is determined cyclically for a current point in time depending on a predefined intake manifold model and free of a temperature measurement value of the gas which is assigned to the current point in time - the model temperature for the current point in time is determined depending on a model temperature, which was determined for a previous point in time, - depending on the model temperature determined for the current point in time, a cylinder air mass is determined which, after the gas exchange valves are closed, is located in the respective cylinder, in which in a second operating state T temperature measurement value of the gas is provided, which is representative of a temperature of the gas at the current time, - a temperature correction value is determined depending on the model temperature for the current time and the temperature measurement value provided, - the temperature correction value is assigned to the intake manifold model and - at least in the first and the second operating state, the model temperature for the current state is determined as a function of the temperature correction value by means of the intake manifold model and in which the model temperature for the current time is adapted as a function of the temperature measurement value provided, by correcting the model temperature in the direction of the temperature measurement value. Method for operating an internal combustion engine comprising an intake tract (1) and one or more cylinders (Z1 to Z4), to which gas inlet valves (12) and gas outlet valves (13) are assigned, wherein gas exchange valves include gas inlet valves (12) and gas outlet valves (13) that in a first operating state - a model temperature of a gas in the intake tract (1) is determined cyclically for a current point in time depending on a predetermined intake manifold model and free of a temperature measurement value of the gas which is assigned to the current point in time, the model temperature for the current point in time is determined as a function of a model temperature which was determined for a previous point in time, - Depending on the model temperature determined for the current point in time, a cylinder air mass is determined which is in the respective cylinder after the gas exchange valves have been closed, in the second operating state a temperature measurement value of the gas is provided which is representative of a temperature of the gas at the current time, a temperature correction value is determined depending on the model temperature for the current time and the temperature measurement value provided, - The temperature correction value is assigned to the intake manifold model and - At least in the first and the second operating state, the model temperature for the current state is determined as a function of the temperature correction value by means of the intake manifold model and in which the model temperature for the current time is adapted as a function of the temperature measurement value provided, by the model temperature as a function of the amount of the difference the model temperature and the provided temperature measurement value are corrected in the direction of the temperature measurement value. Method according to one of the preceding claims, in which a model pressure of a gas in the intake tract (1) is determined cyclically for a current point in time depending on the predefined intake manifold model and free of a pressure measurement value of the gas which is assigned to the current point in time, the model pressure for the current time is determined depending on a model pressure that was determined for a previous time, - The cylinder air mass is determined depending on the model pressure determined for the current point in time. Procedure according to Claim 3 , in which in the second operating state - a pressure measurement value of the gas is provided which is representative of a pressure of the gas at the current time, - a pressure correction value is determined depending on the model pressure for the current time and the pressure measurement value provided, - the pressure correction value the Suction pipe model is assigned and - at least in the first and the second operating state, the model pressure for the current state is determined as a function of the pressure correction value by means of the suction pipe model. Procedure according to Claim 4 , in which in the second operating state - the pressure measurement value of the gas is provided, which is representative of a pressure of the gas at the current time and - the model pressure for the current time is adjusted depending on the pressure measurement value provided. Procedure according to Claim 5 , in which the model pressure for the current time is adjusted depending on the pressure measurement value provided by correcting the model pressure by a predetermined factor in the direction of the pressure measurement value. Procedure according to Claim 5 , in which the model pressure for the current time is adjusted depending on the pressure measurement value provided, by correcting the model pressure depending on the amount of the difference in the model pressure and the pressure measurement value provided in the direction of the pressure measurement value. Device for operating an internal combustion engine comprising an intake tract (1) and one or more cylinders (Z1 to Z4), to which gas inlet valves (12) and gas outlet valves (13) are assigned, wherein gas exchange valves include gas inlet valves (12) and gas outlet valves (13), wherein the device is designed to be a method according to one of the Claims 1 to 7 perform.

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